3. The Neumann Laplacian

Optimizers for Neumann eigenvalues among convex sets of a fixed perimeter
Problem 3.1.
[A. Henrot] Prove the existence of optimizers for $$ \sup\{\mu_k(\Omega)\colon \Omega\subset \mathbb{R}^n \mbox{ convex, }P(\Omega)=P_0\} $$ and $$ \inf\{\mu_k(\Omega)\colon \Omega\subset \mathbb{R}^n \mbox{ convex, }P(\Omega)=P_0\}. $$ 
Maximizing the ratio of Neumann eigenvalues among convex domains
Conjecture 3.2.
[A. Henrot] For any convex $\Omega\subset \mathbb{R}^n$, it holds that $\mu_2(\Omega)/\mu_1(\Omega)\leq 4$.
 $\mu_k(\Omega)/\mu_1(\Omega)\leq k^2$.

Hot spots conjecture
This conjecture is often attributed to J. Rauch, who has formulated it in terms of largetime behaviuor of the heat semigroup generated by the Neumann Laplacian. On the qualitative level, this conjecture manifests as follows: for an insulated flat piece of metal with an arbitrary initial temperature distribution, given enough time, the hottest point on the metal will lie on its boundary.Conjecture 3.3.
The first nontrivial Neumann eigenfunction does not have global extrema in the interior of $\Omega$ if $\Omega\subset \mathbb{R}^2$ is simply connected.
 $\Omega\subset \mathbb{R}^n$ ($n \ge 3$) is convex.

Directional hot spots conjecture
Let $\Omega \subset \mathbb{R}^2$ be a centrally symmetric bounded convex domain and let $u$ denote the eigenfunction corresponding to the first nontrivial Neumann eigenvalue.Problem 3.4.
[T. Beck] Can one always find a direction $\mathbf{e}\in\mathbb{R}^2$ such that \begin{equation} \mathbf{e}\cdot \nabla u >0 \quad \mbox{in }\Omega? \end{equation}
Cite this as: AimPL: Shape optimization with surface interactions, available at http://aimpl.org/shapesurface.